1. A. Rivetti – INFN Sezione di Torino Lecture II Lecture II: Linear circuit theory review Amplifier basics MOS small signal model

2. A. Rivetti – INFN Sezione di Torino Nodal analysis IsIs VsVs R1R1 R2R2 R3R3 R4R4 Nodal analysis provides a systematic and reliable method to calculate all voltages and currents in a linear circuit Nodal analysis

3. A. Rivetti – INFN Sezione di Torino Writing nodal equations R2R2 R4R4 IsIs VsVs R1R1 R3R3 v1v1 v2v2 Nodal analysis

4. A. Rivetti – INFN Sezione di Torino Writing the circuit matrix IsIs VsVs R1R1 R2R2 R3R3 R4R4 v1v1 v2v2 Nodal analysis

5. A. Rivetti – INFN Sezione di Torino Solving the circuit matrix Nodal analysis

6. A. Rivetti – INFN Sezione di Torino Another example IsIs VsVs R1R1 R2R2 R3R3 R4R4 Nodal analysis

7. A. Rivetti – INFN Sezione di Torino Lecture II Lecture II: Linear circuit theory review Amplifier basics MOS small signal model

8. A. Rivetti – INFN Sezione di Torino Amplifier characteristic  The input-output characteristic of an amplifier is usually a non-linear function  Over some interval of the input signal, this function can be approximated by a polynomial:  For narrow range of the input signal, we may write:  The above expression does not obey the superposition principle Amplifier basics

9. A. Rivetti – INFN Sezione di Torino Small signal model  If a 0 does not depend on the signal, we can write:  This is an expression that obeys the superposition principle  The small signal model takes into account only variations of signals within a circuit  The small signal equivalent circuit can be studied with the methods of linear circuit analysis Amplifier basics

10. A. Rivetti – INFN Sezione di Torino Voltage amplifier V s (t) Ri Rs Vout V i (t)  A V = Vout/Vi  Input impedance high (ideally infinite)  Output impedance small (ideally zero) Amplifier basics

11. A. Rivetti – INFN Sezione di Torino VA small signal model AVViAVVi Vout V s (t) RIRI RSRS V i (t) RORO RLRL  Note: impedances may also be complex Amplifier basics

12. A. Rivetti – INFN Sezione di Torino Current amplifier  A V = Iout/Ii  Input impedance small (ideally zero)  Output impedance high (ideally infinite) Ri Rs Iout I i (t) I s (t) Amplifier basics

13. A. Rivetti – INFN Sezione di Torino CA small signal model  Note: impedances may also be complex Amplifier basics Ri Rs I i (t) I s (t) RLRL I out (t) Ro I s (t)

14. A. Rivetti – INFN Sezione di Torino Transconductance amplifier V s (t) Ri Rs Iout V i (t)  A V = Iout/Vi  Input impedance high (ideally infinite)  Output impedance high (ideally infinite)  Important: the gain is not a number Amplifier basics

15. A. Rivetti – INFN Sezione di Torino TCA small signal model  Note: impedances may also be complex Amplifier basics V s (t) RIRI RSRS V i (t) RLRL I out (t) Ro I s (t)

16. A. Rivetti – INFN Sezione di Torino Transimpedance amplifier  A V = Vout/Ii  Input impedance small (ideally zero)  Output impedance small (ideally zero)  Note: Gain is not a number Amplifier basics Ri Rs Vout I i (t) I s (t)

17. A. Rivetti – INFN Sezione di Torino TA small signal model  Note: impedances may also be complex Ri Rs I i (t) I s (t) AVViAVVi Vout RORO RLRL Amplifier basics

18. A. Rivetti – INFN Sezione di Torino Lecture II Lecture II: Linear circuit theory review Amplifier basics MOS small signal model

19. A. Rivetti – INFN Sezione di Torino Simplified small signal DC model RSRS gmgm  V GS  I DS == nn C OX W L (V GS – V TH ) 2 nn C OX W L I DS = The MOS transistor in saturation can be seen as a voltage controlled current source V s (t) gmVs MOS small signal DC model

20. A. Rivetti – INFN Sezione di Torino Practical example What is the equivalent small signal model of this? W=100  m L=10  m  n C OX =190  A/V 2 V TH =0.6 V Vdrain=2.5 V Vgate=1.25 V MOS small signal DC model V gate V drain VsVs

21. A. Rivetti – INFN Sezione di Torino Gm simulation(1) MOS small signal DC model Vs=1mV pk-pk 355.7 356.7 012 time (  S) current (  A)

22. A. Rivetti – INFN Sezione di Torino Gm simulation (2) MOS small signal DC model Vs=250mV pk-pk 355 660 current (  A) 012 time (  S)

23. A. Rivetti – INFN Sezione di Torino Output impedance MOS small signal DC model V gate V drain VsVs r0r0

24. A. Rivetti – INFN Sezione di Torino Including the output impedance RSRS gmgm  V GS  I DS == nn C OX W L (V GS – V TH ) 2 nn C OX W L I DS = ro 1 I DS = The MOS transistor in saturation can be seen as a voltage controlled current source with finite output impedance V s (t) ro gmVs MOS small signal DC model

25. A. Rivetti – INFN Sezione di Torino Bulk transconductance g mb  V BS  I DS == nn C OX W L (V GS – V TH )  V SB  V TH = gmgm  2  F + V SB MOS small signal DC model RSRS V s (t) ro gmVs gmbvbs For a more accurate model, the bulk effect must also be taken into account

26. A. Rivetti – INFN Sezione di Torino Small signal DC model The saturated MOS transistor is a voltage controlled current source with finite output impedance RSRS V s (t) ro gmVsgmbvbs gm models the gate transconductance gmb models the bulk transconductance (the bulk effect) MOS small signal DC model

27. A. Rivetti – INFN Sezione di Torino Some numbers… gmgm  V GS  I DS == 2 nn C OX W L I DS ro 1 I DS = I DS = 100  A, W/L=50,  n C OX =190  A/V 2 =0.01V -1 gm = 1mS ro = 1M  MOS small signal DC model